Lp space

An Lp space consists of measurable functions whose absolute values raised to a power p are integrable, with functions identified when they agree almost everywhere. For p ≥ 1, its standard norm measures their size; for 0 < p < 1, the corresponding size measure is not a norm.

Measure space: A set equipped with a collection of measurable subsets and a measure assigning them sizes. Lp membership depends on integrating over a specified measure space.
Lp norm: For p ≥ 1, the norm of f is (∫|f|^p)^(1/p), with essential supremum used at p = ∞. This formula turns integrability into a measure of function size.
Lp convergence: Convergence in which the Lp norm of the difference from the limit tends to zero. It is the natural notion of approximation measured by the space's size.
L1 space: The space of integrable functions with norm equal to the integral of absolute value. It is the endpoint where integrability measures total absolute mass.
Measurable function: A function whose inverse images of measurable sets are measurable. Only measurable functions are eligible for the integral defining Lp.
Minkowski inequality: The inequality stating that the Lp norm of a sum is at most the sum of the Lp norms. It supplies the triangle inequality that makes Lp a normed space for p ≥ 1.
Lp duality: The representation of continuous linear functionals on Lp by integration against functions in a conjugate space, under suitable conditions. It turns function-space estimates into integral pairings.
L2 space: The space of square-integrable functions with norm given by the square root of their squared integral. Its inner product gives geometric tools unavailable in general Lp spaces.
Lebesgue integral: An integral defined using measure and approximation by simple functions. It gives the integral of the absolute pth power.
Hölder's inequality: A bound on the integral of a product using norms with conjugate exponents. It controls products of functions from paired Lp spaces.

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